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2019 On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow
Tej-eddine Ghoul, Slim Ibrahim, Van Tien Nguyen
Anal. PDE 12(1): 113-187 (2019). DOI: 10.2140/apde.2019.12.113

Abstract

We consider the energy-supercritical harmonic heat flow from d into the d-sphere Sd with d7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one-dimensional semilinear heat equation

t u = r 2 u + ( d 1 ) r r u ( d 1 ) 2 r 2 sin ( 2 u ) .

We construct for this equation a family of C solutions which blow up in finite time via concentration of the universal profile

u ( r , t ) Q ( r λ ( t ) ) ,

where Q is the stationary solution of the equation and the speed is given by the quantized rates

λ ( t ) c u ( T t ) γ , , 2 > γ = γ ( d ) ( 1 , 2 ] .

The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski (Camb. J. Math. 3:4 (2015), 439–617) for the energy supercritical nonlinear Schrödinger equation and by Raphaël and Schweyer (Anal. PDE 7:8 (2014), 1713–1805) for the energy critical harmonic heat flow. Then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed-point theorem. Moreover, our constructed solutions are in fact (1)-codimension stable under perturbations of the initial data. As a consequence, the case =1 corresponds to a stable type II blowup regime.

Citation

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Tej-eddine Ghoul. Slim Ibrahim. Van Tien Nguyen. "On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow." Anal. PDE 12 (1) 113 - 187, 2019. https://doi.org/10.2140/apde.2019.12.113

Information

Received: 23 September 2017; Accepted: 9 April 2018; Published: 2019
First available in Project Euclid: 16 August 2018

zbMATH: 06930185
MathSciNet: MR3842910
Digital Object Identifier: 10.2140/apde.2019.12.113

Subjects:
Primary: 35B40 , 35K50
Secondary: 35K55 , 35K57

Keywords: blowup , Differential geometry , harmonic heat flow , stability

Rights: Copyright © 2019 Mathematical Sciences Publishers

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